Lake Forest College Lake Forest College Publications

Senior Theses Student Publications

4-28-2014 Electromagnetically Induced Transparency and Slow Light Kevin Spotts Lake Forest College, [email protected]

Follow this and additional works at: http://publications.lakeforest.edu/seniortheses Part of the Optics Commons, and the Quantum Physics Commons

Recommended Citation Spotts, Kevin, "Electromagnetically Induced Transparency and Slow Light" (2014). Senior Theses.

This Thesis is brought to you for free and open access by the Student Publications at Lake Forest College Publications. It has been accepted for inclusion in Senior Theses by an authorized administrator of Lake Forest College Publications. For more information, please contact [email protected]. Electromagnetically Induced Transparency and Slow Light

Abstract This thesis describes an experimental observation of electromagnetically induced transparency and ultra-slow light, as well as a description of the theories that describe them. We report electromagnetically induced transparency resonances with ultra-slow group velocities of light of order 400 m/s. We exploited the D1 line of warmed 87Rb vapor with 5 torr of helium buffer gas by establishing a Λ-configuration that employed the triply degenerate magnetic sublevels of the 5S1/2 (F=1) and 5P1/2 (F’=1) hyperfine ts ates. In doing so, we were able to derive our signal and control fields from one laser.

Document Type Thesis

Distinguished Thesis Yes

Degree Name Bachelor of Arts (BA)

Department or Program Physics

First Advisor Michael M. Kash

Second Advisor R. Scott chS appe

Third Advisor Elizabeth W. Fischer

Keywords diode laser quantum optics

Subject Categories Optics | Quantum Physics

This thesis is available at Lake Forest College Publications: http://publications.lakeforest.edu/seniortheses/30 Lake Forest College Archives Your thesis will be deposited in the Lake Forest College Archives and the College’s online digital repository, Lake Forest College Publications. This agreement grants Lake Forest College the non-exclusive right to distribute your thesis to researchers and over the Internet and make it part of the Lake Forest College Publications site. You warrant: • that you have the full power and authority to make this agreement; • that you retain literary property rights (the copyright) to your work. Current U.S. law stipulates that you will retain these rights for your lifetime plus 70 years, at which point your thesis will enter common domain; • that for as long you as you retain literary property rights, no one may sell your thesis without your permission; • that the College will catalog, preserve, and provide access to your thesis; • that the thesis does not infringe any copyright, nor violate any proprietary rights, nor contain any libelous matter, nor invade the privacy of any person or third party; • If you request that your thesis be placed under embargo, approval from your thesis chairperson is required.

By signing below, you indicate that you have read, understand, and agree to the statements above. Printed Name: Kevin Spotts Thesis Title: Electromagnetically Induced Transparency and Slow Light

This thesis is available at Lake Forest College Publications: http://publications.lakeforest.edu/seniortheses/30

LAKE FOREST COLLEGE

Senior Thesis

Electromagnetically Induced Transparency and Slow Light

by

Kevin Spotts

April 28, 2014

The report of the investigation to be undertaken as a Senior Thesis, to carry two courses of credit in the Department of Physics

Michael T. Orr Michael M. Kash, Chairperson Krebs Provost and Dean of Faculty

R. Scott Schappe

Elizabeth W. Fischer

i

ABSTRACT

This thesis describes an experimental observation of electromagnetically induced transparency and ultra-slow light, as well as a description of the theories that describe them. We report electromagnetically induced transparency resonances with ultra-slow group velocities of light of order 400 m/s. We exploited the D1 line of warmed 87Rb vapor with 5 torr of helium buffer gas by establishing a Λ-configuration that employed the triply degenerate magnetic sublevels of the 5S1/2 (F=1) and 5P1/2 (F’=1) hyperfine states. In doing so, we were able to derive our signal and control fields from one laser.

ii

ACKNOWLEDGEMENTS

I owe thanks to many people for all that I have been able to accomplish this year and in the past four years. While it has certainly not been an easy or stress free journey, I enter “the real world” with many true friends and a great supportive family that I can count on for anything. I left home nearly eight years ago on a journey to find myself and my purpose on Earth. Nowhere in my wildest imagination did I expect that this journey would lead me to graduate college with a degree in physics. With that said, special thanks are due to Dr. Kash for his enthusiasm in teaching and advising; it has been a pleasure to be your student and advisee. Just over 15 years ago, Dr. Kash was part of a group of pioneers at Texas A&M that helped pave the way for future research in slow light. Now, with great pleasure, we bring slow light to Lake Forest College.

iii

TABLE OF CONTENTS I. INTRODUCTION ...... 1 II. THE THEORY ...... 2 A. The hydrogen atom...... 3 1. Time-independent perturbation theory ...... 5 2. The nature of light ...... 7 3. Absorption and emission of electromagnetic radiation ...... 10 B. The rubidium atom ...... 12 1. The fine structure of rubidium ...... 13 2. The hyperfine structure of rubidium-87 ...... 15 3. The magnetic sub-levels...... 17 C. Photonics ...... 18 1. Transmission ...... 18 2. Group velocity ...... 23 D. Electromagnetically-induced transparency ...... 27 1. The degenerate Λ-configuration ...... 27 2. A classical model of EIT susceptibility...... 32 E. Ultra-slow light ...... 34 III. THE APPARATUS ...... 35 A. Summary of the experiment ...... 39 B. The laser ...... 40 C. The Reference Arm ...... 44 D. Signal and Control Field Generation ...... 46 1. PCBS and linear polarizer ...... 47 2. Acousto-optic modulator (AOM) ...... 48 3. NPCBS ...... 50 E. Beam Expansion ...... 50 F. Interaction Arm ...... 51 1. Interaction cell ...... 51 2. Signal detection ...... 52 IV. THE EXPERIMENT ...... 53 A. Electromagnetically-induced transparency ...... 53 B. Ultra-slow light ...... 62 V. CONCLUSION ...... 64 A. Further studies ...... 65 VI. REFERENCES ...... 66

1

I. INTRODUCTION

The phenomena of light, has inspired curiosity since the origins of mankind. From the beginning of our existence, the stars that provide an exquisite night sky, and gave us light to see, intrigued our thoughts and studies. However, for most of us, our knowledge of light is confined to a surface level understanding of the physical properties that we can see and feel on a daily basis. For example, we are all certainly aware of the fact that ultraviolet electromagnetic radiation can give some of us a nice bronze skin color, and others—like myself—a nice red sunburn. While the majority of us understand the repercussion of some light phenomena, few understand the fundamental photonic interactions that are occurring. Many of the truly intricate beauties of light occur at the quantum level where light interacts with atoms and molecules.

The most fascinating properties and effects of light—the ones that we physicists get to delve into—are the ones that we cannot see with the human eye, our greatest microscopes or even with a strictly classical physics approach. Light behaves both as a particle (photons) in some instances and as a wave in others. Therefore, our treatment of light requires an approach that includes both the particle like properties and the wave-like properties. With such an examination of light phenomena, we can begin to understand its interactions with atoms that enable quantum mechanical effects to occur. In this thesis, I will undertake coverage of some of the amazing effects of electromagnetic radiation through quantum optics on my way to describing and demonstrating how we can harness the power of light by slowing it down and perhaps even stopping it.

Photons are the fastest carriers of quantum states, but their main strengths are also their weakness because they are very difficult to localize and store. Current quantum 2 optical techniques enable one to control light using atoms and to manipulate the states of atoms using light. This work describes the creation of ultra-slow group velocities of light as part of an expansive effort to develop techniques that can precisely manipulate quantum states for the purpose of developing quantum information systems.

II. THE THEORY

Quantum theory lets a system be placed into a superposition of its distinct states.

Specifically, optical and atomic interference phenomena are central to many areas of quantum optics and atomic physics. Electromagnetically induced transparency (EIT) is one such phenomenon.

EIT is a material dispersion mechanism that significantly alters the index of refraction of a medium within a small frequency range by establishing a superposition of quantum’s states. In this arrangement, quantum interference effectively cancels out the absorption of resonant light, which is then transmitted through the optical medium where it would normally be absorbed. This alteration establishes a dispersive medium that can produce group velocities of light that are orders of magnitude smaller than the typical speed of light in vacuum, c, or 2.998 108 m/s. The group velocity of a wave-front depends on both the index of refraction of the medium through which it propagates and the derivative of the index of refraction with respect to frequency. Before we jump into the theory of EIT and ultra-slow light, it will be useful to have a basic sense of the physics associated with a simple system such as the hydrogen atom and its photonic interactions. Also, a simple understanding of refraction indices and the specific ways that electromagnetic radiation interacts with matter in general will be useful. Once we have 3 developed a firm understanding of photonic interactions and relations, we will move on to the more complex system explored in this research.

A. The hydrogen atom

Atoms are experimentally verified to be made up of discrete energy levels that can host a specific number of electrons that are “free” to transition from one energy level to the next according to the selection rules derived from quantum mechanics. To understand these transitions, we will consider the most fundamental atom and element, hydrogen, as well as review the concepts of emission, transmission and absorption. Once we have a firm understanding of the quantum mechanics behind the discrete energy levels of hydrogen, we can apply quantum mechanical corrections to describe the composition of other atoms—in this case rubidium.

The hydrogen atom is comprised of a single dense proton, essentially motionless and of charge e, and a much lighter orbiting electron with charge –e that is bound by the mutual attraction of opposite charges. According to Coulomb’s law, the potential energy of this arrangement is

e2 1 Vr   . (1) 4 0 r

In 1913, Niels Bohr developed a simple model to describe this system known as the Bohr

Model, which depicts the atom as a small and motionless positively charged nucleus surrounded by a less massive electron that travels in circular orbits around the nucleus like Fig. 1. 4

FIG. 1. The Bohr model of hydrogen. A negatively charged electron confined to an atomic shell orbits the positively charged nucleus with an angular momentum, L. Decay from one orbital to another is accompanied by an emitted photon with energy ∆E = hf , where h is Planks constant and f is the frequency of the emitted photon. The orbits in which the electron may travel are specified by the black rings (n=1, 2, 3 …); their radiuses increase by n2.

Through this model, Bohr successfully approximated the discrete energy levels of the electron orbitals about the hydrogen nucleus based solely on the principle quantum number, n, with the Bohr formula:1

2 2 me1 E1 Enn   ,  1, 2, 3,  (2) 242 nn 2 2 0 where E1 = -13.61 eV is the ground state energy (n  1). Although Bohr’s model was a great first approximation, it was eventually corrected by the more accurate, and never- the-less more intricate theory of quantum mechanics and Schrödinger’s equation:2

2   2  V   ih , (3) 2mt 5 published in 1924.

1. Time-independent perturbation theory

Using quantum mechanics and the time-independent form of Schrödinger’s equation in three dimensions

2  2 VE    , (4) 2m or more simply

HEn n n , (5) we can account for other effects taking place within an atom through the application of

Time Independent Perturbation Theory, which treats each additional interaction as a slight perturbation of the original Bohr model. A perturbation to the original system yields a new Hamiltonian

HHH0 ' (6)

where H0 represents the Hamiltonian of the original unperturbed system and H ' represents the Hamiltonian of the perturbed system. This new Hamiltonian gives corrections to the original energies

1 0 0 EHn  n', n (7) which says that the first-order correction to the energy is the expectation value of the perturbation, in the unperturbed state. The new total energies will be given by

01 EEEn n  n ... (8)

Although in many cases the energy corrections are relatively small in magnitude to the

Bohr energies,i we will find it useful to account for them in order to exploit intrinsic finer

i A detailed list characterizing the relative magnitude of each energy correction can be found in Introduction to Quantum Mechanics, by David Griffiths. 6 structures of atoms. For example, the motion of the electron around the proton gives a magnetic field in the rest frame of the electron. That magnetic field interacts with the intrinsic dipole moment of the electron according to the Hamiltonian3

Hso ' μBe  (9) where B is the aforementioned magnetic field and 𝝁풆 is the dipole moment of the electron. This combines with the Hamiltonian associated with the relativistic energy of the electron

pˆ 4 H '  (10) r 8mc32 to form the perturbation responsible for the fine structure of hydrogen

HHHfs'''. r so (11)

The hydrogen atom also has a hyperfine structure. The hyperfine structure, often referred to as spin-spin coupling, is caused by the magnetic field that is created from the dipole moment of the proton due to is intrinsic spin. This magnetic field interacts with

the intrinsic dipole moment of the electron. Effectively, the spin of the proton S p , is

coupled to the spin of the electron, Se , to create total spin that is conserved: SSSep.

The Hamiltonian has the same form as Eq. (9). Most introductory textbooks on quantum mechanics discuss these perturbations along with a description of the physical cause, the relevant first-order Hamiltonian, the good quantum numbers, and the first order energy correction. However, for now it shall suffice to say that atoms exhibit energy levels reflecting fine and hyperfine structure.

In this research, we work exclusively with a gas of rubidium atoms transitioning between the ground and excited states of the hyperfine structure. Therefore, I will eventually move into a detailed analysis of the rubidium atom, but first I will review the 7 ideas behind electron transitions—quantum jumps—between states through the absorption and emission of electromagnetic radiation.

2. The nature of light

An understanding of the properties that underlie the absorption and emission of electromagnetic radiation is crucial in nearly any apparatus involving light and atoms.

The structure of atoms is highly intricate, so once we account for all of the corrections described by quantum mechanics, we will need a mechanism by which to confirm our understanding. For example, according to the Zeeman Effect, atoms subjected to an external magnetic field will experience a shift in their discrete energy levels, and consequently the frequencies that drive those transitions shift slightly as well. The characteristics of these changes are often best investigated through the processes of absorption and emission of electromagnetic radiation. In this research, I work extensively with light from a laser, which is absorbed and re-emitted from atoms contained in a vessel called a cell. Therefore, some things should be said about electromagnetic radiation and its photonic interactions with matter.

8

FIG. 2. An electromagnetic wave. The polarization of the wave is determined by the motion of the electric field by convention—in this case the light is linearly polarized.4

An electromagnetic wave consists of transverse oscillating electric and magnetic fields. Examining Fig. 2, we notice that the electric field, the magnetic field and the direction of propagation are all mutually orthogonal. The Poynting vector specifies the energy per unit time per unit area in the direction of propagation of the electromagnetic wave:

1 SEB (12) 0

The Lorentz force:

FLorentz  q( E  v  B ), (13) states that a charged particle, q, moving with velocity, v, in the presence of an electromagnetic field will experience a force. The relative magnitude of the force caused by the electric field component is significantly larger than the relative magnitude of the force caused by the magnetic field component, which we can confirm by looking at the relationship between the magnetic field and the electric field5

1 B kˆ E. (14) c 9

This says that the magnetic field component is a factor of 108 smaller than the electric field component because kˆ is the unit vector that describes the direction of propagation of the light. Therefore, a charged particle—such as an electron bound to an atom—will move predominantly along the direction of the electric field. This provides us with a valid reason to choose a convention that mainly concerns the electric field. With that said, the polarization of light is described by the behavior of the vector amplitude of the electric field component, which may be doing a number of different things.

If the wavelength of the light considered is long (nanometers) in comparison to the size of the atom (picometers), then we can ignore special variation in the field, and the atom is exposed to an oscillating electric field

ˆ E  E0 cos wL t k , (15) which can potentially drive an electron transition into an excited state or back down to a ground state based on the electric fields polarization. The states that may be coupled by the electric field are specified by the selection rules derived through quantum mechanics.

These selection rules depend on the polarization of the light.

Linear polarization is characterized by the oscillation of the vector electric field amplitude in one plane, as depicted in Fig. 2. In our experiment, the creation of two beams of orthogonally polarized light—vertically and horizantally polarized as I later refer to them—enabled us to split a single laser beam into two separate beams.

The aforementioned fields could then be converted into coherent counterrotating fields of left and right circular polarization, which were used in my research to drive

87 specific transitions associated with the magnetic sublevels, mF , of the Rb hyperfine structure. 10

Circular polarization is characterized by the rotation of a constant vector electric field amplitude as is depicted in Fig. 3. The distinction between right and left circulatly polarized light is arrived at by a simple convention. Envision the circularly polarized light wave to be traveling towards, the observer. If the electric field rotates in a clockwise fashion then the light is said to be right-circularly polarized, whereas if it where to be rotating counter-clockwise then the light is said to be left-circularly polarized.6

FIG. 3. Circularly polarized light as defined by an arbitrary axis. The electric field component of the light wave rotates in a circular fashion as it propagates.

3. Absorption and emission of electromagnetic radiation

When an electron is temporarily excited to a higher state within an atom, either through thermal excitation or by the application of resonant incident electromagnetic radiation, eventually the electron makes a transition downward or decays to a lower energy state (often the ground state) through the release of excess energy in the form of a photon—a quanta of energy.ii The energy of the photon corresponds to the difference

ii If we want to allow for atomic transitions through the emission and absorption of radiation in our quantum mechanical treatment of an atom, then we must resort to Time-Dependent Perturbation Theory because the potential of atoms can vary in time. Time-Dependent Perturbation Theory exploits the idea that in the event that the time- dependent portion of the Hamiltonian is small in comparison to the time-independent part, then it can be treated as a perturbation. The application of Time-Dependent Perturbation Theory to the emission and absorption of radiation is described in detail in Introduction to Quantum Mechanics, Second Edition, by David Griffiths, p. 314.

11 between the initial and final states involved in the resonant transition, which for hydrogen is7

11 EEE   13.61 eV  . (16)  if 22 nnif

According to the Planck Formula,8 the energy of the photon is proportional to its frequency:

hc E  . (17)  0 

This is also the characteristic energy and frequency that incident electromagnetic radiation would need in order to drive the transition.

Looking once again at the hydrogen atom, by combining Eq. (16) and Eq. (17) we can derive the Rydberg formula for the spectrum of hydrogen,

1 1 1 EER    , (18) nnfi22  nnfi where R is the Rydberg constant and has the numerical value9

2 2 EE11 me 71 R   3  1.097  10 m . (19) hc2 c 4  c  4 0

Unfortunately, making use of this for atoms other than hydrogen is quite difficult.

As it turns out there is no exact solution to the complex Hamiltonian of more dense atoms:10

ZZ2 2 2 2 1 Ze 1  1  e H   j ψ      . (20) 2mr 4 2 4 j1 00 j   j k rrjk

Therefore, we must resort to more elaborate approximation methods in our treatment of

12 complicated atoms. Fortunately in this research we are concerned with the behavior of rubidium atoms, which can be approximated as a hydrogen atom for reasons to be discussed.

B. The rubidium atom

Rubidium (N 37) is an alkali metal that can be found in the first column of the periodic table along with hydrogen, lithium, sodium, potassium, cesium and francium. All of these atoms have some things in common; most importantly in this case, they all have one valence electron in their outer most shell. This enables us to approximate the more dense alkalis as hydrogen-like atoms. For the heavier atoms down the column, such as rubidium, the charge of the electrons in the filled shells can be approximated as a shield that screens out the excess charge of the additional protons in the nucleus and thus the single, outermost valence electron does not feel the full extent of the excess charge. As

[11] states, “Angular momentum tends to throw the electron outward, and the farther out it gets, the more effectively the inner electrons screen the nucleus (roughly speaking, the inner most electron “sees” the full nuclear charge Ze, but the outermost electron sees an effective charge hardly greater than e).”11

Some other properties of elementary rubidium are that it is highly reactive and oxidizes rapidly in air. Natural rubidium is a mixture of two isotopes: 85Rb and 87Rb.

Rubidium-85 is stable and makes up about 72.17% of all the natural mixture, while 87Rb

12 is slightly radioactive (τ1/2 = 49 billion years) and comprises only 27.83% of the mixture.

Some of the most important properties of rubidium exploited in my research are that it is easily vaporized near room temperature, it has a convenient electromagnetic absorption range that makes it a good target for laser manipulation of atomic states, and as 13 mentioned before it has a single valence electron in its outer most shell.

The ground state configuration of the 37 electrons in rubidium is

1s2 2 s 2 2 p 6 3 s 2 3 p 6 4 s 2 3 d 10 4 p 6 5 s 1 , (21) according to the general rules for filling orbitals.13 The first 36 electrons form a stable arrangement that contributes no net angular momentum to the system. Thus the sole valence electron that resides in the 5S orbital (L = 0)—the ground state configuration— accounts for the angular momentum of the system and we shall center our attention there.

When excited to the 5P orbital (L = 1), the valence electron soon decays back down to the ground state, and gives off a photon characterized by the change in energy. The specific energy can be determined with some examination of the fine and hyperfine structure of rubidium.

1. The fine structure of rubidium

The fine structure is a result of the coupling between the orbital angular momentum L of the outer electron and its spin angular momentum S. These two components combine to form the total electronic angular momentum

JLS, (22) which must lie in the range

LSJLS    . (23)

The rubidium 5P orbital splits into two distinct sub-levels with slightly lower energy due to the intrinsic relativistic correction and spin-orbit coupling, whereas the 5S orbital is only affected by the relativistic correction as depicted in Fig. 4.iii The two

iii If the reader is not familiar with the terms fine structure and hyperfine structure, now would be an ideal time to resort to any quantum mechanics textbook where they are most certainly described in more detail. 14

2 2 2 transitions to consider are referred to as the D1 line (5 S1/2 5 P1/2) and D2 line (5 S1/2

2 iv  5 P3/2).

FIG. 4. The fine structure of 87Rb. The relativistic and spin-orbit corrections combine to form the fine structure correction.

The D1 line has transition energy of 0 1.560 eV, so resonant transitions from the D1 line will correspond to photons with wavelengths of   794.979 nm .14 The D2

line has transition energy of 0 1.589 eV, so resonant transitions from the D2 line will correspond to photons with wavelengths of   780.241 nm. 15 The wavelength difference between photons associated with the D1 and D2 transitions is about 14.734 nm, which makes it very difficult to drive both transitions with the same laser. The research that I conducted solely focused on the D1 line of 87Rb, so I will concentration my analysis of the hyperfine structure on the D1 line.

iv The notation used here describes the state of the atom: , n is the orbital, S is the spin, J is the electronic angular momentum and L is the orbital angular momentum, where

15

2. The hyperfine structure of rubidium-87

The hyperfine structure accounts for the interaction between the nuclear magnetic spin and the intrinsic dipole moment due to the spin of the electron. In other words, the hyperfine structure is a result of the coupling of the total electronic angular momentum J with the total nuclear angular momentum I to form the total atomic angular momentum,

F.

FJI, (24) which must be in the range

JIFJI    . (25)

Rubidum-87 has an experimentally determined nuclear magnetic spin number

3 1 I  and for the D1 line the total electronic angular momentum is J  . We can 2 2 determine the possible hyperfine structure sublevels by determining all of the possible

2 2 values of F at each fine structure level as illustrated in Fig. 5. Both the 5 S1/2 and 5 P1/2 states have possible F values of 1 or 2, therefore the fine structure splits into two distinct hyperfine levels. 16

FIG. 5. Hyperfine structure of the 87Rb D1 line along with associated wavelengths, frequencies, wavenumbers, energies and g-factors.16 17

3. The magnetic sub-levels

Orbitals Hyperfine Levels Magnetic Sublevels

F =2 2 5 P1/2

F =1

f e d c

F =2 2 5 S1/2

F =1

FIG. 6. The hyperfine structure of 87Rb with intrinsic magnetic sublevels. The selection rules mandate that transitions may only occur if The available transition are color coded and labeled f, e, d and c correspondingly following P. Siddons et al.17

The hyperfine energy levels are also comprised of innate magnetic sublevels, mF.

The values of these sublevels must be within the range

F  mF  F. (26)

The F = 2 states consist of six magnetic sublevels mF   2,  1, 0, 1, 2 , while the F

= 1 states only contain three mF  1, 0, 1 . Figure 6 depicts the available magnetic sublevels and allowed transitions.

The selection rules for circularly polarized light mandate that transitions between the magnetic sublevels of the hyperfine structure of 87Rb obey

mF  1. (27)

We find this information useful in choosing the polarization of our laser fields because different polarizations drive specific transitions associated with an atom. In the following 18 sections I will address some of the theory behind the interaction of electromagnetic radiation with atomic states, as well as how we can measure the interaction.

C. Photonics

Photonics is the term broadly used to contain concepts relevant to the generation, transmission, modulation, amplification and detection of light. These areas have found ever-increasing applications in optics, signal processing, sensing, and computing and all of them have applicability to the research that I have conducted. In the following sections, I will draw my attention to some of the photonics that must be understood in order to experiment with complex electromagnetic phenomena such as electromagnetically induced transparency and ultra-slow light.

1. Transmission

The transmission of electromagnetic waves through an atomic gas like 87Rb is governed by the refractive index of the gaseous medium and the absorption coefficient, or collectively the susceptibility of the medium. Therefore, we should briefly explore some characteristics about the index of refraction and the susceptibility of a medium subject to propagating electromagnetic radiation.

When a monochromatic electromagnetic wave propagates through a medium, the ratio of the speed of light in vacuum to that in matter is known as the index of refraction, n:18

c n  , (28) v where c is the speed of light in vacuum, and v is the phase velocity of the light in the medium. If we quickly rearrange Eq. (28) for v, then we have an expression for the phase 19 velocity of the wave through the medium,

c v  . (29) phase n

We can also write Eq. (28) in terms of the relative permittivity and relative permeability of the medium to be considered19

 n rr (30) 00 where 휖푟 is the relative permittivity and 𝜇푟 is the relative permeability of the optical

20 medium. For most materials 𝜇 is very close to 𝜇0, so we can reduce Eq. (30) to

n  r (31)

The relative permittivity and permeability can also be written in terms of susceptibility

rm1 (32)

re1, (33) where 휒푒 is the electric susceptibility and 휒𝑚 is the magnetic susceptibility. Looking once again at the phase velocity of the light in matter, we can rewrite Eq. (29) in terms of Eq.

(31) and Eq. (33):

cc vphase  . (34) re1

Since 휖푟 is almost always greater than 1, light indeed propagates slightly slower in matter than it does in vacuum.

Problems arise in defining the refraction index of a medium because the relative permittivity 휖푟 depends on the frequency of propagating light. As is well known in optics, the refractive index of a medium is a function of wavelength or frequency. This dependence is called dispersion. In addition, whenever the speed of a wave depends on 20 its frequency, the supporting medium is called dispersive.21 There are a few types of dispersion to be considered in what we call a dispersive medium subjected to frequency varying electromagnetic radiation: normal dispersion and anomalous dispersion.

Normal dispersion arises within a medium when the frequency of incident light,

ω, is detuned far from any atomic resonant frequency, ω0. In this arrangement an increasing radiation frequency leads to an increasing refractive index

dn  0. (35) d

As a result the phase velocity is reduced from its vacuum value, c. Therefore, electromagnetic waves travel with a slower phase velocity (Fig. 7). For example, a prism or a raindrop bends blue light more sharply than red, and is responsible for spreading white light out into a rainbow of colors because of normal dispersion.

The refractive index of an optical medium behaves much differently near an atomic resonance. As [22] says, “The index of refraction of an optical medium can reach values as high as 10 or 100 at frequencies near an atomic resonance. The price that must be paid for such high dispersion is usually an accompanying high absorption.”22 This phenomenon is known as anomalous dispersion,v which supervenes when the frequency of incident radiation, ω, is within the vicinity of a resonant atomic absorption frequency,

ω0. In this arrangement the index of refraction is characterized by a sharp decrease as the frequency of radiation increases (Fig. 7):

dn  0, (36) d and the phase velocity is therefore suddenly increased.

v Anomalous means irregular. We label the dispersion mechanisms near a resonance as “irregular” because most often we would not find electromagnetic waves propagating through a medium for which they are resonant. 21

( 0

FIG. 7. Index of refraction versus frequency (Upper Graph) of an optical medium near a resonant atomic absorption (Lower Graph). The horizontal axis is the frequency detuning of the laser source, where ω0 is the resonant frequency, ω is the laser frequency, and γ is the absorption coefficient.

In the previous section we saw that the propagation of electromagnetic waves through most matter is governed by intrinsic properties of the material such as the frequency dependence of permittivity. In this section, we will account for the frequency dependence of the permittivity, , and susceptibility using a simplified classical model for the behavior of electrons in dielectrics subject to electromagnetic radiation. 22

Consider the atoms that make up a dispersive medium to be N damped harmonic oscillators. If we picture each electron as being attached to the end of an imaginary

spring, with spring constant kspring , as is shown in Fig. 8, then we can write the binding force as23

2 Fbinding  k spring#1 x   m e 0 x, (37)

where x is the displacement from equilibrium, me is the electrons mass, and 0 is the natural oscillation frequency,

kspring #1 0  . (38) me

x

E1 v electron z kspring#1

FIG. 8. A classical model of an electron, mass me, on a spring.

The damping force in this case is made to be proportional yet opposite in direction to the velocity

dx Fm  , (39) damping e dt 23 where  is the coefficient of absorption. In the presence of an electromagnetic wave of frequency , polarized in the x direction like Fig. 8, the electron is subjected to a driving force

Fdriving  qE qE0 cos( t ). (40)

In this case, q is the charge of the electron and E0 is the amplitude of the electric field at point z where the electron is situated. If we put all of this into Newton’s second law then we get

d2 x dx m m  m 2  qEcos(  t ), (41) edt2 e dt e 00 a damped driven harmonic oscillator. In this case, a resonance frequency—the “favorite frequency of the electrons”24—leads to large amplitudes that correspond to strong absorption and therefore less transmission. Conversely, smaller amplitudes coincide with less absorption and more transmission.

2. Group velocity

Since waves of different frequencies travel at different phase velocities in a dispersive medium, a wave form that incorporates a range of frequencies—such as amplitude modulated light—will change shape as it propagates, with each frequency component traveling at a frequency dependent phase velocity,25

 v  , (42) phase k where ω is the frequency of radiation and k is the wavenumber:

2n k . (43)  c

24

We can also talk about the packet of light frequencies as a whole—the “envelope” as I shall call it— which travels at the so-called group velocity26

d v  . (44) group dk

If we assume a simple dispersion relation

 kn ( ), (45) medium c

then we can rewrite our group velocity as27

c v  . (46) group dn n  d

Evidently the group velocity depends on both the index of refraction and the derivative of the index of refraction with respect to frequency.

To further appreciate the idea of a group velocity let’s considered the superposition of two electromagnetic waves defined by their electric field and with slightly different frequencies, ω1 and ω2, as well as wavenumbers, k1 and k2:

Wave #1: E1 E0cos( k 1 z 1 t ) x (47)

Wave #2: E2E 0cos( k 2 z 2 t ) x , (48)

12 . (49)

No matter the scenario, the superposition of the two travelling waves and their corresponding phase velocities creates a group velocity like that portrayed in the snap shot of Fig. 9. The behavior of that group velocity will be the topic of interest in this section and many of the sections to follow:

 12 vgroup  . (50) k k12 k 25

ω1 < ω2

vphase1 vphase2 vgroup

FIG. 9. Snap shot of two travelling waves and their group envelope. The two sinusoidal waveforms have slightly different frequencies ω1 (Purple) and ω2 (Green) and corresponding phase velocities vphase1 and vphase2, which combine to form the group envelope (Blue) with group velocity, vgroup.

Examining Fig. 9, when the waveforms are out of phase a destructive interference is established that corresponds to small amplitudes in the group envelope, whereas when the waveforms are in phase a constructive interference is produced that corresponds to maximum amplitudes in the group envelope. To decipher the characteristic behaviors of the group velocity associated with our two travelling waves we must consider a few different scenarios.

The least interesting case occurs when our two travelling waves are in vacuum. In this case the two waves will propagate at the same speed and thus the group velocity is

vgroup v phase12  v phase  c . (51)

By far the more interesting scenarios arise when our waves are brought to propagate in a dispersive medium. Recalling the dispersion relation illustrated in Fig. 7, let’s consider a 26 couple of scenarios in which the behavior of the two waves and their composite group velocity will behave differently.

First, let’s consider the case where both waveforms are tuned near a resonance and influenced by anomalous dispersion. In this circumstance Wave #1 will propagate

slightly slower than Wave #2 ()vvphase12 phase according to Eq. (29), and the group velocity will slightly exceed the speed of light in vacuum:

vcgroup  . (52)

Yes, it is possible for the group velocity to exceed the speed of light and without breaking the laws of physics I might add.

Second, let’s consider that both ω1 and ω2 are far detuned from any atomic resonances; let’s say they are tuned to the leftmost normal dispersion portion of the dispersion relation in Fig. 7. In this case as well, the waves will travel through the medium at slightly different speeds, once again according to Eq. (29). Wave #1 will

propagate slightly faster than Wave #2 (vvphase12 phase ), and the group velocity will be slightly slower than the speed of light in vacuum:

vcgroup  . (53)

This same effect would be achieved if we tuned the frequencies of Wave #1 and Wave #2 to the right-most normal dispersion portion of Fig. 7.

The big result here is that we have determined that under the proper circumstances, i.e., when the index of refraction varies positively with respect to frequency, the group velocity of light can be slower than the speed of light in vacuum.

This leads us to the climactic moment of this thesis, where I shall move forward in 27 explaining electromagnetically induced transparency and ultra-slow light, which can produce group velocities of light that are several orders of magnitude smaller than the speed of light in vacuum.

vcgroup (54)

D. Electromagnetically-induced transparency

As touched upon earlier, the strength of the interaction between light and an ensemble of atoms or molecules is a function of the wavelength or frequency of light.

When the light frequency matches the resonant driving frequency 0 of a particular atomic transition the optical response of the medium is significantly enhanced and light propagation is then accompanied by strong absorption and dispersion mechanisms.28 Electromagnetically induced transparency is a technique that can be used to make a resonant, opaque optical medium transparent by evolution of quantum interference between a three level atomic system that cancels the absorption of light and thereby creates a very steep, positively varying dispersion relation.

dn  0. (55) d

1. The degenerate Λ-configuration

A lambda configuration is a model of an ideal, three-state atomic system that can be used to describe EIT. In practice, a lambda configuration could be achieved by using different states of the hyperfine structure, as is done in the work of many other EIT experiments.29 In this research we use the degenerate magnetic sublevels of the 87Rb hyperfine structure; therefore, I shall refer to our arrangement as the degenerate lambda configuration. 28

| ⟩

Ωc Ωs

| ⟩ | ⟩

FIG. 10. The degenerate Λ-configuration. The excited state | ⟩ is coupled to two degenerate ground states | ⟩ and | ⟩ (as might be established in the magnetic sublevels of an atom’s hyperfine structure) by a signal field with Rabi frequency Ωs and a driving field with Rabi frequency Ωc. The absorbing states driven simultaneously causes a quantum interference that cancels the absorption of the light and thus the atom is transparent to the resonant radiation and transmission increases.

Referring to Fig. 10, two degenerate ground states are coupled to the same excited state by two separate fields, which I will refer to as signal (subscript s) and control

(subscript c) fields following the notation used by Lukin et al.30 The fields have Rabi frequencies Ω푐 and Ω푠, which are constructed to simultaneously drive transitions between ba and bc correspondingly. In this arrangement a destructive interference is established where no absorption takes place. This can be measured as an increase in transmission. The Rabi frequencies are defined as

dEab c Ω c  (56) and

dEac s Ωs  . (57)

Here 풅푎푏 and 풅푎푐 are the transition dipole moments from state ab and ac correspondingly, and 푬푠 and 푬푐 are the vector electric field amplitudes of the signal and control fields, which includes their respective polarizations. The dot product 29 includes a factor of cos 휃, where 휃 is the angle between the polarization of light and the transition dipole moment. When the two are parallel or antiparallel the interaction is the strongest, and when they are orthogonal there is no interaction at all. For now, I will disregard the polarizations of the two fields until I have defined a real atomic system.

In the arrangement of our degenerate lambda configuration, atoms are effectively made transparent to the resonant Rabi frequencies. Atoms in this transparent state are said to be coherently trapped in a “dark” state. The Hamiltonian for this system, in the rotating-wave approximation, is obtained by a subtle generalization of the Hamiltonian for a two-level atom interacting with a single-mode field to the three-level arrangement with a two-mode field by:31

ˆ H ΩΩΩcb a  s c a Ω, ca b  s a c (58) which can be rewritten as

Ω푐 Ω푠 ̂ √Ω푐 Ω푠 | ⟩ | ⟩ ⟨ |

√ √ { Ω푐 Ω푠 Ω푐 Ω푠 } (59)

Ω푐 Ω푠 √Ω푐 Ω푠 | ⟩ ⟨ | ⟨ |

√ √ { Ω푐 Ω푠 Ω푐 Ω푠 }

By introducing the coupling constant

Ω √Ω푐 Ω푠 (60) we can simplify the notation of the Hamiltonian to 30

Ω푐 Ω푠 Ω푐 Ω푠 ̂ Ω { | ⟩ | ⟩} ⟨ | Ω| ⟩ { ⟨ | ⟨ |} (61) Ω Ω Ω Ω

If we define

Ω Ω |푩⟩ ≡ 푐 | ⟩ 푠 | ⟩ (62) Ω Ω as the “bright” state, then the Hamiltonian takes on the form

̂ Ω|푩⟩⟨ | Ω| ⟩⟨푩| (63)

The state that is orthogonal to |푩⟩ is

Ω Ω |푫⟩ ≡ 푐 | ⟩ 푠 | ⟩ (64) Ω Ω the “dark” state. These newly defined “dark” and “bright” states form a complete new basis set where |푩⟩ is a superposition of the original ground states and |푫⟩ is the state where the atoms that make up our resonant, dispersive medium no longer absorb. In this newly defined system of states, the process of coherent population trapping into the

“dark” state is much more obvious, so we can redraw the Λ-configuration.

| ⟩

Ω Γ

|푩⟩ |푫⟩

FIG. 11. The Λ-configuration in terms of “bright” and “dark” states. In this case |푩⟩ couples with | ⟩ where transitions from |푩⟩ | ⟩ are driven by the coupling constant . Atoms in state |푩⟩ are excited to | ⟩ they eventually spontaneously decay by the decay constant Γ into one of the ground states. Eventually all of the atoms will end up in |푫⟩ where the atoms are said to be in a dark state and can no longer absorb radiation. 31

When the atoms that make up the dispersive medium are trapped in the “dark” state |푫⟩ by EIT, the dispersion picture that we saw in Fig. 7 is significantly altered. Since atoms in the “dark” state no longer absorb resonant light, our dispersion and absorption picture needs to be reworked. Figure 12 accounts for both the steep positive variation in the index of refraction with respect to frequency (“non-anomalous” dispersion) and the induced transparency associated with EIT. Since atoms coherently trapped in the “dark” state are decoupled from resonant radiation in an ideal EIT medium, the susceptibility vanishes and the refractive index becomes equal to 1 (vacuum). This means that the phase velocities of resonant monochromatic wave fronts will be equivalent to their vacuum values, c. 32

( 0

FIG. 12. Index of refraction versus frequency (Upper Graph) of an optical medium near a resonant atomic absorption in the presence of EIT (Lower Graph). The horizontal axis is the frequency detuning of the laser source, where ω0 is the resonant frequency, ω is the laser frequency, and γ is the absorption coefficient.

2. A classical model of EIT susceptibility

Recall our earlier model of an electron on a spring (Fig. 8) in which a single driving force was applied to our electron-spring model. When the frequency of the driving force E approached the resonant frequency of the system (Eq. 38), large 33 amplitudes corresponded to strong absorption in our simplified atomic susceptibility model. In the case of EIT, we must introduce a second spring and driving force into our model to represent our Λ-configuration (Fig. 13).

FIG. 13. A Classical model of EIT. The driving forces of two electric fields with opposite polarization (exactly out of phase) and equivalent resonant frequency cancel the movement of the electron.

As before, large amplitudes of oscillations correlate to strong absorption.

However, in the case of EIT, the simultaneous driving of the electron by the two equivalent electric fields with opposite polarization prevents the electron from oscillating at all and there is no amplitude. Based on this and our definition of absorption in this classical model, we say that the absorption is canceled and the electron is transparent to the resonant driving frequency. This is the classical approach to explaining EIT. In reality atoms are much more complicated then electrons on springs and must be described quantum mechanically. Never-the-less, whether we view EIT classically or quantum mechanically is beside the point. This research is mainly concerned with the results.

When an ensemble of atoms is subjected to EIT, the resulting non-anomalous dispersion significantly affects how light interacts with the medium.

34

E. Ultra-slow light

It was proposed in 1991 by Harris et al. that it was possible to manipulate the group velocity by taking advantage of the resulting refractive properties in EIT media.32

By examining the resultant dispersion relation, we can begin to imagine exactly how we might construct group envelopes of light that would feel the full effect of the EIT dispersion model established by the narrow transparency window. In reality the transparency resonance is much smaller and occurs over a much narrower range of frequencies than Fig. 12 would lead one to believe, but for convenience we show a widened version. The nature of our new dispersion curve would greatly affect light with frequency slightly above or below the resonant frequency.

Recalling our earlier dispersion model, light tuned to the normally dispersive frequencies of a medium where there is a moderate positive variation in index of refraction resulted in slow group velocities. In an EIT media that variation is much larger. As a result, an envelope of carefully evolved frequencies can be made to move with a group velocity that is much slower than what we saw in our normally dispersive medium. This is the essence of ultra-slow light. In next section of this thesis, I will provide a detailed analysis of the apparatus that I constructed in order to establish group envelopes of light that both maintained an EIT resonant medium and reduced the group velocity of a signal pulse by several orders of magnitude to merely 400 m/s.

35

III. THE APPARATUS

The work of those before me made lengthy efforts to explain some of the components of the apparatus in great detail and therefore at times I shall refer to the work of my predecessors. My intention is to cover only those components that were added by me and crucial in seeing ultra-slow light. The following figures reveal a detailed diagram of the apparatus and comprehensive illustrations of all of the instrumentation and the corresponding connections.

Figure 1, shows that the diagram of the apparatus is divided into perforated, colorful boxes: The Laser, The Reference Arm, Signal and Control Field Generation,

Beam Expansion and finally the The Interaction Arm. These divisions strategically break up the apparatus into parts with components that coincide in purpose and will form the key subjects to be explored within the apparatus. 36

10 3 - Lens (PD1) Photo 2 1 Detector 1 Mirror Lens Lens Mirror > - > - Iris 9 /4 Plate/4 1 NPCBS λ (Signal (Signalfield) Zehnder - (Control field) (Control Shielding Magnetic TwoLayer Arrangement Mach Polarized - Polarized HP - PCBS 1 PCBS Beam Beam Expansion VP Control Control 87 & Buffer & 87 - GasVapor (Insidecell) Rb and Generation Field Mirror Linear Signal Polarizer (SolenoidInside) Holder House 87 Vapor 87 - Neutral Special LFC Reference Cell (Insidecell) Photodector Rb DensityFilter Interaction Interaction Arm /4 Plate/4 2 λ 2 Holder/heater InteractionCell (InsideShielding) ) Ω Beam 2 Splitter Heaters Resistor PCBS (5 W, (550 1 Optical Blocker Mirror 4 - Beam Splitter Lens (PD2) Photo Detector 2 Pair Prism 100 mm) 100 - 300 mm) 300 The The Laser (f= f= 50.20 mm) 50.20 Analyzer Spectrum Optical Isolator = 30.10 mm) 30.10 = f= f ReferenceArm Concave Concave ( Convex Lenses - - Convex ( Convex ( Convex - - Plano Plano Bi : : Bi Fiber Optic 1: 1: Coupler Lens 2: Lens 3 Lens 4: Lens 4 5 6 1 3 7 8 2 FIG. 14. A detailed diagram of the apparatus used to see slow light. The corresponding components of the apparatus are grouped into sections: The Laser, Reference Arm, Signal and Control Field Generation, Beam Expansion, and Interaction Arm. 37

2 Radiation Mate Mate Channel: Function PD 1 2 EXT 2 1 LFC Generator Design 7 House Oscilliscope 6 Output Electromagnetic - Optic Fiber Coupler Analyzer Spectrum Spectrum 2 1 5 ECDL Optic Optic - Cable o 4 BNC Fiber MHV MHV TE Output to PZT to to LD Output LD 3 PZT V Output Output t V to to PZT H MHV )

Ω Mod In Mod Control Control Voltage Labs Labs Labs = 1.1 M 1.1 = TOT

Current Current

LFC Laser Laser LFC Thor Thor (R PZT Driver PZT Controller Controller

Control Box Control Wavemeter High Voltage High Temperature Temperature

+ V V In Bias Input to 15 Control BNC External MHV MHV

OP445: OP445: V Control Control Voltage Out Sync 1 V 15 - 85 V) 85 - Lock Box Lock Channel: (~70 Agilent DC DC Agilent Generator +15 V/ +15 LFC SRS Function Function SRS Oscilliscope 1 2 3 EXT 3 2 1 Power Supply Power Supply Power Function Sweep

FIG. 15. A detailed diagram of the instrumentation and relevant connections involved in constructing both the Signal and Control Field Generation and the Interaction Arm sections of the apparatus.

38

3 2 Thor Thor Channel: 1 2 EXT 2 1 Labs PD Labs Oscilliscope 8 cell inside cell Output Sync DC PS DC Solenoid shielding with shielding Generator or or Generator Pasco Function Function Pasco interaction Wrapped magnetic in Wrapped Generator 10 SRS Function Function SRS Radiation PD PD 1 Thor Function Labs Labs Input Video Video Electromagnetic 9 Zehnder Zehnder AOM - AOM RF RF AOM Generator Arrangment Mach RF Output Output RF (40 MHz)

FIG. 16. A detailed diagram of the instrumentation and relevant connections involved in constructing both the Laser and Reference sections of the apparatus.

Radiation 39

A. Summary of the experiment

In attempt to produce ultra-slow group velocities of light, two precisely tuned 795 nm (infrared) laser beams derived from a single source are arranged to propagate through a warm atomic ensemble of 87Rb gas and 5 torr of He buffer gas. To begin, coherent, linearly polarized light from the laser source—in this case an external-cavity diode laser

(ECDL)—propagates through the laser section of the apparatus. Here the laser light is converted into linearly polarized light at 45 degrees to the vertical by means of an optical isolator and then the cross-sectional area of the beam is reshaped from elliptical to circular by an anamorphic prism pair. This gives the output beam an approximately symmetric Gaussian distribution of intensity.

Upon exiting the laser portion of the apparatus the beam is split up into various paths using 2 beam splitters. Some of radiation is reflected into a reference arm where the stability, frequency and approximate wavelength are carefully monitored by a spectrum analyzer, a reference cell, and a wavemeter. The remaining radiation is transmitted through the beam splitters and manipulated into signal and control fields in the Mach-Zehnder Arrangement.

The Mach-Zehnder arrangement separates the light into two distinct (yet still coherent) fields of horizontal and vertical polarization with a polarizing cube beam splitter (PCBS1). While separated, an acousto-optic modulator amplitude modulates the signal field into an envelope of frequencies. Then a non-polarizing cube beam splitter

(NPCBS) recombines the beams. Following the Mach-Zehnder arrangement the laser beam is expanded to approximately 5 mm in diameter and monitored by means of photo- detector (PD1). The diameter of the beam can be adjusted by implementation of an iris or changing the lenses used in the expansion process. 40

After expansion, the signal and control fields are converted into left and right circular polarization respectively by means of a quarter wave plate before being directed through the interaction arm. The interaction arm is host to the interaction cell, which sits in the middle of a long solenoid that is wrapped in two layers of magnetic shielding. The interaction cell is filled with a trace 87Rb and 5 torr of He buffer gas. Upon exiting the interaction cell, transmitted light is converted back into its original components of linearly polarized light (signal and control field) by means of a second quarter wave plate. The signal field is then separated from the control field with a second polarizing cube beam splitter (PCBS2) and monitored by a photo detector (PD2) for any delays with respect to PD1.

B. The laser

A laser is a device that emits light through a process of optical amplification, which is based on the stimulated emission of electromagnetic radiation. Although of lately the use of “laser” has taken on a form of its own, it originated as an acronym,

L.A.S.E.R., which stands for light amplification by stimulated emission of radiation. The advent of the laser was a tremendous achievement in the field of science because of its wide applicability in the study of various quantum effects.

In our apparatus, we used an external cavity diode laser (ECDL) arrangement accompanied with an optical isolator and an anamorphic prism pair in order to establish a highly stable, tunable laser source with a symmetric output beam and a long lifetime. The laser diode is a Sacher Lasertechnik product (SN: AD 144462) originally specified at 830 nm (infrared), but brought down to ~795 nm by the dedicated work of David Curie, LFC class of 2013. 41

At 795 nm the laser diode is operating in a range associated with the D1 line of

87Rb, which is at the edge of the diodes spectral width. Therefore, we are unable to achieve maximum output power (200 mW). With that said, lack of power never became an issue and we readily had about 13 mW at our disposal for the entire apparatus. In what follows I describe the external cavity diode laser arrangement.

1. External cavity diode laser (ECDL) arrangement

GRATING

FIG. 17. A modified tunable external-cavity diode laser arrangement. A piezo stack enables the incident angle θi of the diffraction grating to be finely tuned so that the desired wavelength may be selected.33

An ECDL arrangement is a useful lasing method that makes use of a laser diode chip, typically with only one end antireflection coated, and a diffraction grating in order to establish the lasing cavity. Operating a laser diode in an extended cavity provides frequency-selective feedback and is a very effective method of reducing the laser’s line- width as well as improving its tunability. The arrangement also enables the laser diode output to be tuned over a few nanometers in a single-mode with good precision. There are two popular external cavity arrangements, namely the Littman-Metcalf configuration and 42 the Littrow configuration. These two configurations (Fig. 18) differ based on which order of reflection from the diffraction grating is used to establish the laser cavity. The wavelength of reflected light behaves in accordance with the diffraction law for reflective gratings,

s 휃 s 휃𝑚 (65)

In the Littrow configuration this simplifies to

md 2 (sinb ) (66)

My EDCL is a modified Littrow configuration (Fig. 17) that followed the work of

Hawthorn et al.34 The addition of a single plane mirror that is held fixed relative to the diffraction grating with a simple mount attached rigidly to the arm of the existing laser keeps the output beam direction constant. The benefit of this is that slight tuning of the wavelength does not call for total optical realignment.

FIG. 18. In (a) the Littrow configuration, the first order diffraction from the grating is fed back into the laser diode establishing the laser cavity, and the 0th order reflected light forms the output beam. In (b) the Littman-Metcalf configuration, higher orders of reflection off of the diffraction grating are used to establish the laser cavity.35

Some important factors that control the frequency of emitted light from the laser including the temperature of the diode, the current supplied to the diode and the angle of incident radiation on the diffraction grating. These things were controlled using a 43

ThorLabs TEC 2000 Temperature Controller, a ThorLabs Laser Diode Controller 50, a high voltage amplifier (Doug Rank, LFC Class of 2008) and a Stanford Research

Systems DS345 Function Generator. The SRS Function Generator output a triangle wave with a definable amplitude and frequency that was sent into the high voltage amplifier control box. The high voltage amplifier amplified the waveform and was used to control the outputs of the PZT stack and the ThorLabs Current Controller. The thesis of Dawson

Nordurft, LFC Class of 2010, covers these control mechanisms in more detail.

The stability of the laser can be furthered improved by implementing an optical isolator. Essentially an optical isolator prevents any unwanted feedback from getting back to the laser source and establishing competing laser cavities. Details of the inner workings of an optical isolator are explored by Nordurft as well.

One characteristic of laser diodes is that their outputs have an elliptical output. To create a more circularly symmetric Gaussian profile, I implemented the use of an anamorphic prism pair. Anamorphic prism pairs work by magnifying an elliptical beam in one dimension as illustrated in Fig. 19. The benefits of a circular cross-section are yet to be fully explored and perhaps need to be better understood. 44

FIG. 19. The magnification process of an anamorphic prism pair.36

C. The Reference Arm

The reference arm of the apparatus consists of a spectrum analyzer, a wavemeter, and a reference cell monitored by a photo detector. The purpose of the reference arm is to provide feedback about how the laser is operating by measuring the frequency, absorption by a reference cell, and the approximate wavelength of the output beam. The reference arm played a significant role in enabling me to finely tune the laser output to a precise frequency that would optimize EIT and slow light.

1. Reference cell:

My main modification to the reference arm was the implementation of an isotropic 87Rb reference cell. The previously reference cell contained both naturally occurring isotopes of Rubidium, 85Rb and 87Rb. The main issue with this arrangement is that the absorption spectrums overlap with one another in some cases. This makes it difficult if not impossible to distinguish between the associated spectrums. Since this 45 research works strictly with 87Rb transitions, it made sense to transition to an isotropic arrangement.

With that said, the importance of the reference cell cannot be overstated. I monitored the transmission of radiation propagating through the reference cell using a photodiode, which provide crucial feedback about the performance of our laser and enabled me to see the atomic spectrum of 87Rb on an oscilloscope. By looking at the DC levels of the photodiode and the components controlling laser frequency I could precisely tune the laser to the resonant absorption frequency of the desired transition. A typical scan of a stable, well-tuned laser on a given day is represented in Fig. 20. Each dip in transmission on channel 1 (yellow) corresponds to a particular atomic absorption governed by a particular resonant frequency.

The broadness associated with the absorptions is due to the Doppler Effect.

Within the atomic absorption cell the vaporized rubidium atoms are moving around. In the most extreme case, some of the atoms are moving toward the direction of propagation and others are moving away. Due to this, the atoms and their relative motions interact with the resonant electromagnetic radiation differently. Atoms moving in the same direction as the propagating light experience a redshift and atoms moving opposite to that of the direction of propagation experience a blue shift. 46

FIG. 20. A scan of the ECDL tuned in the vicinity of the D1 line of 87Rb, as monitored on an oscilloscope. A dip in transmission corresponds to the absorption of resonant electromagnetic radiation produced by the EDCL. Channel 1 monitors the transmission through the reference cell as monitored by a photodiode. Channel 2 monitors the DC level of the control voltage input to our laser control box. Channel 3 monitors the DC level of the voltage output to the diffraction grating PZT stack. The frequency of the laser is increasing from left to right.

D. Signal and Control Field Generation

The generation of both the signal and control field had a huge magnitude of impact on the success of the experiment. The goal was to generate two coherent fields with perpendicular polarizations and variable power that could eventually be turned into counter-propagating left and right circular polarization. To achieve this I used a Mach-

Zehnder arrangement as depicted in Fig.21. The use of a linear polarizer, a polarizing cube beam splitter (PCBS1), an acousto-optic modulator (AOM), mirrors, a non- polarizing cube beam splitter (NPCBS), and irises eventually gave rise to the two desired fields. 47

HP: Signal Field

VP: Control Field

FIG. 21. Signal and control field generation in the Mach-Zehnder Arrangement.

1. PCBS and linear polarizer

Horizontally polarized light

Vertically polarized light

FIG. 22. The Polarizing Cube Beam Splitter37

The polarizing cube beam splitter works by splitting the incident light into two distinct beams with orthogonal polarization. The transmitted light is horizontally polarized, and the reflected light is vertically polarized. In my arrangement, the signal field was horizontally polarized and the control field was vertically polarized. A linear polarizer was used to adjust the relative powers of both polarizations. With the 48 transmission axis of the linear polarizer at 45 degrees from vertical, equal amounts of power in both the horizontally and vertically polarized beams exited the PCBS. However, that did not correlate to equal amounts of power following the non-polarizing cube beam splitter. The angle of linear polarizer was adjusted for the desired power. For example, in my slow light experiments the signal field was set to be about 10% of the control field.

2. Acousto-optic modulator (AOM)

Deflected beam (Shifted 40 MHz)

Carrier beam

FIG. 23. The acousto-optic modulator38

Acousto-optic modulators (AOMs) are devices that enable the frequency, intensity, and direction of a laser beam to be modulated. Inside an AOM, incoming light diffracts off acoustic wave fronts that propagate through a crystal such as glass. Sound waves traveling through a crystal can be modelled as crests of increased refractive index alternating with troughs of decreased refractive index. Modulation of the incoming light is achieved by varying the amplitude and frequency of the acoustic waves travelling through the crystal.

In my experiment I use an AOM to amplitude modulate the signal beam by about

50% into an envelope of frequencies. By adjusting the alignment, light entering the AOM is diffracted into a first order 40MHz frequency shifted beam that is prevented from 49 further propagation by an iris. Next, we arrange for the AOMs radio frequency generator to be smoothly turned off and on by means of a function generator. As a result the power of the incident signal field is shifted between the deflected and carrier beam. This process functions both as a process of amplitude modulation and as the generation of envelope signal pulses, which can be monitored by a photodiode on an oscilloscope.

FIG. 24. Acousto-optic modulator cartoon. A transducer generates a sound wave, at which a light beam is partially diffracted. The diffraction angle is exaggerated.39

To create ultra-slow group velocities of light the signal beam is amplitude modulated into an envelope of frequencies that will interact with the EIT medium uniquely and according to our dispersion models from earlier. The modulation process can be thought to have constructed two sidebands of frequencies, one slightly above and one slightly below the original unmodulated light, or the carrier. If the carrier is initially tuned to a resonant frequency necessary to establish an EIT medium with the control field, then the modulation process will yield two side bands that experience the full effects of the extremely steep and positively varying dispersion curve of Fig. 12. Based on the rate that the carrier is amplitude modulated, the frequencies of the sidebands will have associated frequencies. For example, if the signal beam in this experiment is amplitude modulated by 50% at 1 kHz, then the side bands will be constructed of light that is 500 Hz above and below the initial unmodulated signal field. The resonant carrier 50 field will also survive the modulation process if we do not amplitude modulate by 100%.

This will maintain an EIT resonant medium. In theory we should be able to modulate our signal beam faster and observe our side bands become tuned to the anomalous dispersion relations of Fig 12.

3. NPCBS

A non-polarizing cube beam splitter is used in our Mach-Zehnder arrangement to recombine the signal and control fields in an overlapping fashion. It is extremely important to establish good overlap so that the interaction cross-section is maximized in the interaction cell. For example, Fig. 25 depicts the effect of poor overlap and good overlap. In theory this sounds like a simple task but it could take hours to achieve great alignment.

FIG. 25. A demonstration of good/bad signal and control beam overlap. When the beams do not overlap well, the atomic interaction cross-section (bright red) is significantly decreased, whereas when the beams are well overlap the interaction cross-section is large. In order to achieve an optimal EIT, the beams need to be in good optical alignment and significantly overlapped.

E. Beam Expansion 51

The beam expansion section of the apparatus enables me to control the diameter of the beam. I increased the diameter of the beam from 2 mm to about 5 mm by using a telescope. By adjusting the focal lengths of the lenses and the distance between them any number of beam sizes can be achieved. Because the ultimate goal of this research is to make educated calculations about observed group delays, the addition of variable control over the interaction cross-section seemed fitting.

F. Interaction Arm

The interaction arm is the final destination for the signal and control fields. In the interaction arm the two fields are converted in to counter-rotating left and right circular polarization by a quarter-wave plate. Soon after, the fields enter the interaction cell where they interact with a warmed 87Rb gas (~ 75C ) that establishes a dispersive medium.

The light will either be transmitted or absorbed depending on the frequencies of the fields. Transmitted light is converted once again into its linearly polarized components by a second quarter-wave plate. Next the signal and control components are separated once again by a polarizing cube beam splitter. Finally, the signal field is monitored on an oscilloscope by means of a photodiode for any delays with respect to a reference photodiode situated before the interaction cell. With that said, I have made many modifications to the interaction arm of the apparatus since taking over.

1. Interaction cell

By far one of the greatest improvements to the apparatus was the addition of a buffer gas in the interaction cell. Introducing a buffer gas significantly increases the interaction time between the atoms and the propagating fields by restricting their motion to a small area. This restriction also limits decoherences caused from collisions with the 52 walls of the cell. A great model for this that was brought to mind by Dr. Kash is as follows. Envision the idea of a buffer gas as a bowl of rice and peas. Within a bowl there is a lot of rice and fewer peas. The large population of rice restricts the movement of the larger peas about the bowl. In this case, the peas model rubidium atoms and the rice helium atoms for the absorption cell that was used in my slow light experiments.

2. Signal detection

Another dramatic improvement to the apparatus was the installation of a quarter wave plate and polarizing cube beam splitter after the interaction cell. The combination of both separates the signal and control fields once again in order to be monitored separately—this was something that prior attempts at seeing slow light at Lake Forest

College had failed to do and without a doubt contributed to a lack of results.

53

IV. THE EXPERIMENT

A. Electromagnetically-induced transparency

Before I could begin looking for ultra-slow light I needed to ensure that I was achieving the best EIT resonances possible with the apparatus. This required relentless adjusting of optical alignment and exploration of different absorption cells. Ideally, I am looking for an EIT resonance that has a very narrow line-width (kHz) and a high relative transmission. The following figure illustrates an ideal EIT resonance with 100% transmission in a typical spectral absorption as well as the magnitude of the resonances that our apparatus achieved.

1

0.2

Transmission

ω o ωlaser

FIG. 26. EIT resonances. The solid black line represents the normal absorption line without EIT. The green line represents an ideal transparency resonance with 100% transmission. The purple line represents the order magnitude of transparency resonances that we saw with our current apparatus,  20%

I have successfully established EIT on the D1 line of 87Rb by exploiting the

22 5SFPF1/2  1  5 1/2  1 transition associated with the hyperfine structure. This is 54 the simplest transition. I did so by forming a Λ-configuration that employed the magnetic sublevels of the hyperfine structure as in Fig. 27.

Orbitals Hyperfine Levels Magnetic Sublevels

2 5 P1/2 F =1

+ - e σ σ

2 5 S1/2 F =1

FIG. 27. Λ-configuration used to establish EIT and ultra-slow light.

Transitions from | ⟩ | ⟩ were driven by the control field with right circularly polarized light, while transitions from |

⟩ | ⟩ were driven by the signal field with left circularly polarized light, . With this arrangement I was able produce EIT line-widths in the range of 250 kHz and relative transparencies of about 30%. In what follows I have included an example a typically observed EIT resonance, as well as the calculation of the line-width and the relative transmission. Ultra-slow light is highly dependent on the result of EIT; therefore it is very important to understand how well we can establish EIT with our apparatus.

55

EIT established in Isotropic 87Rb & 5 torr He buffer gas (50 mm cell):

After a lot of research on what kind of cells others were using in both EIT and slow light experiments, we determined that using a cell that contained a buffer gas was the way to go. The following figure is a typical resonance that was achieved using a heated (~ 75C ) 50 mm cell containing almost purely isotropic 87Rb and 5 torr of Helium buffer gas. The power of both the control and signal field were set to be approximately

1.5 mW.

0.35 divisions

3 divisions

10 divisions

FIG. 28. Observation of EIT using the FF1  '  1 hyperfine transition of the 87Rb D1 line. Channel 1 is a measure of the transmission through the interaction cell. Channel 2 measures the voltage across a sense resistor in series with the solenoid.

56

To begin the calculation of the line-width we must first extract all of the useful information from the oscilloscope. Examining Fig. 28, I can calculate the rate of change of the voltage into the solenoidvi by

s o s o s (67) s o s s s o

Since channel two represents the voltage across a sense resistor in series with the solenoid, we can apply Ohm’s Law to get the current through the resistor and the solenoid. Because the resistance is constant, the rate of change of the current through the solenoid can be written as

s (68) s Ω

As we know from electricity and magnetism, the current through a solenoid is directly related to the magnetic field it creates. A linear relationship between the magnetic field strength and the amount of current through the solenoid was established by measuring the magnetic field at the center of the solenoid over a range of currents.

vi NOTE: To observe EIT, a DC voltage ramp is applied to the solenoid in order to generate a sweeping magnetic field. When the field is zero, EIT is established. When a field is present the EIT is destroyed due to the shifting of magnetic sublevels in accordance with the week field Zeeman Effect. 57

Magnetic Field of Solenoid vs Current (@Center) 2.5 Equation y = a + b*x Weight No Weighting Residual 5.94603E- Sum of 4 2.0 Squares Pearson's r 0.99998 Adj. R-Squa 0.99997 Value Standard Er Intercept 0 -- 1.5 B Slope 1.7424 0.00276

1.0

(mT)

z B 0.5

0.0

-0.5 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Current (A)

FIG. 29. Magnetic field vs Current for the solenoid. The linear relationship suggests that the rate of change of the magnetic field with respect to current is 1.74 mT/A, or 17.4 G/A. Using the linear relationship established in Fig. 29 we can calculate the magnetic field in the solenoid with some simple arithmetic

(69)

If Eq. (68) and Eq. (69) are combined then the rate of change of the magnetic field can be approximated as

s s (70)

Next, by relating the first order perturbation caused by the Zeeman Effect 58

𝜇 푒 (71) to Plank’s Formula

0 (72) the first order energy correction can be expressed in terms of frequency

𝜇 (73) 푒

As a rate, Eq. (73) can be written as

𝜇 푒 (74)

where for the 512 SF state used to establish EIT, , and is 1.400 1/2  

MHz/G. Now the EIT line width can be obtained from the relation

(75)

To calculate the line-width of the EIT illustrated in Fig. 28, the last piece of information necessary is the temporal width of the transmission resonance at full width half maximum.

s s o s s (76) s o

Plugging in the necessary values the line-width of the EIT is 59

𝜇 푒

(77)

( ) ( ) ( ) s s

Line-width: 푬

Next, to determine the approximate relative transmission associated with our EIT resonance, we must determine the amount of transmission on and off resonance. To do so

I set the laser to be on resonance and recorded the DC levels. Next I set the laser to be off resonance and again took note of the DC levels. The difference between the DC levels on and off resonance will give us a good estimate of the total available transmission of the absorption line. By combining the information of the following figures we can reason the relative transmission of our resonant transparency.

FIG. 30. DC level transmission of the signal field with the laser detuned far from resonance. 60

4.2 divisions

0.4 divisions

FIG. 31. DC level transmission of the signal field with the laser tuned to resonance. The red dashed line represents the DC level off resonance as determined from Fig. 30.

Resonance Transmission:

Total Available Transmission:

Relative Transmission:

NOTE: This represents only one of many cases measured. The typical range for the relative transmissions that were observed was between 10% and 20%.

Analysis:

As was expected, the observed relative EIT transmissions were not as great as the ideal case of 100% transmission. While I observed a relative transmissions range of 10-

20%, in order to achieve relative transmissions of 20% it took relentless optical aligning of the entire apparatus. According to one source, “for typical conditions in warm atoms, that provide a reasonable approximation of the three-level Λ system, EIT-induced transparency > 50% is readily accomplished.”40 Since we did not achieve greater than 61

50% transparency, perhaps it would be useful to further explore how to achieve greater relative transmissions.

In all cases of observed EIT resonances, the peak of the resonance did not occur exactly where the magnetic field was zero. This leads us to believe that even with our two layer magnetic shield some magnetic fields are still present. This is definitely worth further investigation. We believe this external field might be a result of our cell heaters, which are resistive heaters. It may be worth applying a static field to the solenoid to see if the effect can be reduced.

Another important measurement to be analyzed is the line-width of our EIT resonance. For the purposes of slow light experiments, we desired a narrow line-width because a narrow line-width translates to a steeper non-anomalous dispersion relation.

The line-widths of the EIT resonances that I measured were typically in the range of 250 kHz. This is a significantly lower value than the 760 kHz line-width measured by David

Curie, LFC Class of 2013, who employed the same Λ configuration, but used a 75 mm cell filled with natural rubidium.

It appears the addition of a buffer gas significantly improved our EIT. By introducing a few torr of inert buffer gas, the interaction time between the atoms and the field is significantly increased. With that said, since the intention of my research was to produce slow light, I took these EIT results to be suitable conditions for the achievement of slow light.

62

B. Ultra-slow light

I created an optimized EIT medium by employing the use of a 50 mm (~2 inch) absorption cell that contained almost purely isotropic 87Rb and 5 torr He buffer gas. The power of the signal field has been reduced to about 15% of the control field and is amplitude modulated by ~ 50% by applying a sine wave to an AOM. If the width of the signal pulse lies within the EIT spectral window, then the pulse will be delayed, with only modest absorption while propagating through the atomic medium of length L by a the group delay41

tD = L / v group. (78)

The following figure is an example of an observed signal delay:

~ 125μs

FIG. 32. Image of a delayed signal pulse amplitude modulated at 600 kHz. Channel 1 is the reference signal from before the interaction cell. Channel 2 is the delayed signal pulse as measured after the interaction cell. In this arrangement I saw group delays of up to 125 μs. By rearranging Eq. (78) the group velocity can be determined to be… 63

푣𝑔푟표𝑢푝 𝐷 μs

푐 That is !!! 0000 The fractional delay can be calculated by taking the ratio between the group delay and the temporal width of the pulse:

μs 𝐷 푝𝑢 푠푒 μs

As this was the first time that slow light had been observed at Lake Forest

College, it was imperative to ensure that the signals that we observed were not a misinterpretation of some other phenomena occurring. In order to confirm the results, several tests were performed. First, by detuning the laser far from any atomic resonances

EIT should be lost and the group delays should vanish; they did. Next, if we destroy EIT conditions by blocking the control field with an optical blocker, then the delays should be wiped out; they were. Lastly, if we remove the cell, or let it cool, then the observed delays should go away, and they did. The following figure shows what our signal looked like while we were performing these tests: 64

FIG. 33. When the laser is detuned far from any resonances the group delay in the signal pulse goes away completely and we see the two overlapped waveforms. . Channel 1 is the reference signal from before the interaction cell. Channel 2 is the no longer delayed signal pulse as measured after the interaction cell.

The delay in the signal pulses that I observed stood up to many tests and therefore we have verified that ultra-slow light has been observed.

V. CONCLUSION

By the addition of many new controllable variables, I have constructed an apparatus that optimizes electromagnetically induced transparency for the use of slow light experiments. Therefore, I am able to take advantage of the dispersion properties established by an EIT medium in order to create a signal pulse with an ultra-slow group velocity of merely 400 m/s.

65

A. Further studies

The future of ultra-slow light at Lake Forest College is bright. The additions that

I have made to the apparatus introduce many controllable variables that future experimenters will be able to explore. For example, my successors might explore the effects of different beam sizes, modulation rates, field intensities, cell temperatures, or perhaps even different buffer gases. My most recent endeavors explored the power of the control beam. By decreasing the power of the control beam, the power broadened EIT resonance should become narrower and result in even bigger group delays like in the experiments performed by Kash et al.42 In theory, if the control beam is adiabatically reduced to zero—say by the employment of another AOM—then the light will be stopped and trapped in the medium. The pulse can then be retrieved by a subsequent pulse a short time after. However, as my time is coming to an end, and as much as it pains me to says, the future of ultra-slow light at Lake Forest College lies in the hands of those that follow me.

66

VI. REFERENCES

1 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, Upper Saddle River, NJ, 2005), Equation 4.70, p.145.

2 Ibid. Equation 1.1, p. 1. 3 Ibid. Equation 6.58, p. 271. 4 National Weather Service. JetStream – Online School for Weather. “Introduction,” Remote Sensing, 5 Jan 2010, (March 20, 2014).

5 D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Upper Saddle River, NJ: Pearson Addison Wesley, 1999), Eq. 9.49 6 E. Hecht, Optics, 3rd ed. (San Francisco, CA: Addison Wesley, 2002), p. 328 7 Griffiths, Equation 4.91, p. 158. 8 Ibid. Equation 4.92, p. 158. 9 Ibid. Equation 4.94, p. 158. 10 Ibid. Equation 5.24, p. 211. 11 Ibid. p. 215 12 D. A. Steck, Rubidium 87 D Line Data, available online at http://steck.us/alkalidata (revision 2.0.1, 2 May 2008), p.16-17.

13 R. Gautreau, W. Savin, The Theory and Problems of Modern Physics, 2nd ed. (McGraw-Hill, 1999), p. 140. 14 Steck, p. 26. 15 Ibid. p. 26. 16 Ibid. p. 26. 17 P. Siddons, C.S. Adams, C. Ge, and I. G. Hughes. Absolute absorption on rubidium d lines: comparison between theory and experiment. Journal of Physics B: Atomic Molecular and Optical Physics, 41 (155004), July 2008. 18 Griffiths, Equation 9.68, p. 383. 19 Ibid. Equation 9.69, p. 383. 20 Ibid. Equation 9.70, p. 383. 21 Ibid. p.398. 22 M. O. Scully, M. S. Zubairy, Quantum Optics, (United Kingdom: Cambridge University Press, 1997), p. 236. 67

23 Griffiths, Introduction to Electrodynamics, Equation 9.151, p. 399 24 Ibid. p. 400 25 Ibid. Equation 9.149, p. 399. 26 Ibid. Equation 9.150, p. 9.150. 27 G. Welch, private communication 28 Scully, p.225-235. 29 E. Figueroa, F Vewinger, J. Appel, A. I. Lvovsky, On decoherence of electromagnetically- induced transparency in atomic vapor. Institute for Quantum Information Science, University of Calgary, Alberta, Feb. 1, 2008. 30 M. D. Lukin, Colloquium: Trapping and manipulating photon states in atomic ensembles, Reviews of Modern Physics, Volume 75, April 2003. 31 Y. Rostovtsev, University of North Texas, Private Communication 32 Lukin, p. 459 33 R. Scholten, “Optics group: Atom optics,” tech. rep., University of Melbourne, School of Physics, 2006. 34 C. J. Hawthorn, K. P. Weber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum., vol. 72, 2001, p. 4477-4479. 35 D. Curie, Thesis, Lake Forest College (2013), p. 11. 36 ThorLabs. Anamorphic Prism Pair, Picture, .(March, 23, 2014). 37 Newport. Laser Line Polarizing Cube Beamsplitters. (March, 20, 2014). 38 IntraAction Corp. Picture, (March, 20 2014). 39 RP Photonics. Acousto-optic Modulators, (March, 21, 2014). 40 I. Novikova, R. L. Walsworth, Y. Xiao, EIT-based slow light and stored light in warm atoms, Laser and Photonics Reviews, (2011). 41 Ibid. 42 M. M. Kash, et al, 1999 Phys. Rev. Lett. 82, 5229.